Archive for the ‘General Algebra’ Category

On his journey north, Basho stopped at Matsushima and was spellbound by its beauty. The town itself was small, but the bay is studded with some 260 tiny islands. The white stone of the islands has been eaten away by the sea, leaving a multitude of endlessly different shapes, pillars and arches, all crowned with pine trees. Each island is different and unique, and each, with its sculpted white cliffs tufted with pine, is beautiful. It is, however, the whole bay, the combined diversity, that ultimately makes Matsushima one of the three great scenic locations in Japan. So far on our journey we have been admiring the beauty of the islands; the subtlety and intricacy of the different algebras that arise from different patterns, different symmetries. It is time to step back and begin to admire the whole — and in so doing gain a deeper and richer perspective on all the sights of our journey so far.

Numbers are remarkably tricky. We tend not to notice because we live in a world that is immersed in a sea of numbers. We see and deal with numbers all the time, to the point where most basic manipulations seem simple and obvious. It was not always this way of course. In times past anything much beyond counting on fingers was the domain of the educated few. If I ask you what half of 60 is, you’ll tell me 30 straight away; if I ask you to stop and think about how you know that to be true you’ll have to think a little more, and start to realise that there is a significant amount of learning there; learning that you now take for granted. Almost everyone uses numbers regularly every day in our current society, be it through money, weights and measures, times of day, or in the course of their work. Through this constant exposure and use we’ve come to instinctively manipulate numbers without having to even think about it anymore (in much the same way that you no longer have to sound out words letter by letter to read). That means that when we meet a new abstraction, like the symmetries discussed in Shifting Patterns, it seems comparatively complex and unnatural. In reality the algebra of symmetries is in many ways just as natural as the algebra of numbers, we just lack experience. Thus, the only way forward is to look at more examples, and see how they might apply to the world around us.

How hot the sun glows,
Pretending not to notice
An autumn wind blows!*

— Matsuo Basho

What is a haiku? Or, more specifically, what makes a particular composition a haiku, as opposed to one of the many other poetic forms? The defining feature most people will be familiar with is the 5-7-5 syllable structure. Within that basic structure, of course, the possibilities are almost endless, and this is what makes haiku so tantalizing to write: you can shift the words and syllables around to craft your message, and as long as you retain the classic 5-7-5 syllable structure you can still call your work a haiku**.

This is not an isolated trait. We constantly define, and categorise, and classify, according to patterns. We determine a basic pattern, an underlying structure, and then classify anything consistent with that structure accordingly. This is our natural talent for abstraction at work again, seeking underlying patterns and structure, and mentally grouping together everything that possesses that structure. It is the means by which we partition and cope with the chaotic diversity of the world. And yet, despite our natural talent for this, it wasn’t until the last couple of centuries that we had any treatment for this sort of abstraction comparable to our use of numbers to formalise quantity.

As a mathematician there is a story I hear a lot. It tends to come up whenever I tell someone what I do for the first time, and they admit that they don’t really like, or aren’t very good at, mathematics. In almost every case, if I bother to ask (and these days I usually do), I find that the person, once upon a time, was good at and liked mathematics, but somewhere along the way they had a bad teacher, or struck a subject they couldn’t grasp at first, and fell a bit behind. From that point on their experiences of mathematics is a tale of woe: because mathematics piles layer upon layer, if you fall behind then you find yourself in a never ending game of catch-up, chasing a horizon that you never seem to reach; that can be very dispiriting and depressing. In the previous entries we have dealt with subjects (abstraction in general, and the abstraction of numbers) that most people have a natural intuitive grasp of, even if the details, once exposed, prove to be more complex than most people give them credit for. It is time to start looking at subjects that often prove to be early stumbling blocks for some people: fractions and algebra.